
Case 1 \(c\ne 0\) :
In this case,
\(v_1=-c\epsilon _2+b\epsilon _3, \ v_2=(b^2+c^2)\epsilon _1-ab\epsilon _2-ac\epsilon _3\) 
gives a orthogonal frame
field of \(M\) .
Direct computations show the following
formulae:
\(g[1](R(v_1,v_2)v_1,\mathbf {n})=8ac^{2}(b^2+c^2),\ \ g[1](R(v_1,v_2)v_2,\mathbf {n})=8bc(b^2+c^2).\) 
Take a principal frame
\(\lbrace e_1,e_2\rbrace \) . Then
\(\lbrace e_1,e_2\rbrace \)  is expressed as
\(e_1=\cos \mu \frac{v_1}{|v_1|}+\sin \mu \frac{v_2}{|v_2|},\ \ e_2=-\sin \mu \frac{v_1}{|v_1|}+\cos \mu \frac{v_2}{|v_2|}.\) 
Then we have
\(R_{1213}=\frac{8c(b^2+c^2)}{|v_1||v_2|}\left(\frac{ac}{|v_1|}\cos \mu +\frac{b}{|v_2|}\sin \mu \right),\) 
\(R_{2123}=\frac{8c(b^2+c^2)}{|v_1||v_2|}\left(\frac{ac}{|v_1|}\sin \mu -\frac{b}{|v_2|}\cos \mu \right).\) 
From these we have \(\tau ^{\mathcal {V}}(\psi )=0\) 
if and only if \(a=b=0\) .
Hence \(\theta ^3=-\eta \) .
Namley \(M\)  is an integral surface
of the distribution \(\eta =0\) ,
but this is impossible,
since
\(\eta \)  is contact.
(See p. 36, Theorem
in [1]}).


Case 2 \(c=0\) :
Since \(a^2+b^2=1\) , we may write
\(a=\cos \phi ,\ b=\sin \phi \) .
In this case \(u_1=\sin \phi \epsilon _1-\cos \phi \epsilon _2,\ u_2=\epsilon _3\) 
are orthonormal and tangent to \(M\) .
The unit normal \({\mathbf {n}}\)  is given by
\({\mathbf {n}}=\cos \phi \epsilon _1+\sin \phi \epsilon _2\) .
Then we have
\(R(u_1,u_2)u_1=-\sin ^{2} \phi \epsilon _3,\ \ R(u_2,u_1)u_2=-\sin \phi \epsilon _1+\cos \phi \epsilon _2.\) 
Let us denote by \(\mu \)  the angle
between the principal frame
\(\lbrace e_1,e_2\rbrace \)  and \(\lbrace u_1, u_2 \rbrace \) ,
i.e.,
\(e_1=\cos \mu \:u_1+\sin \mu \:u_2,\ \ e_2=-\sin \mu \:u_1+\cos \mu \:u_2.\) 
Using (REF ) and (REF ),
we have
\(R_{1213}=R_{2123}=0\) .
Thus \(\tau ^{\mathcal {V}}(\psi )=0\) 
is fulfilled automatically for
\(M\)  with \(c=0\) .
We have shown in
[2]} that
constant mean curvature surfaces with
\(c=0\)  are Hopf cylinder of constant
mean curvature.
See the proof of Thereom in [2]}.
Furthermore,
the second fundamental form
\({\mathrm {I}}\!{\mathrm {I}}\)  of \(M\)  relative to \({\mathbf {n}}\)  is
given by (cf. (5) and (8) in [2]})
\({\mathrm {I}}\!{\mathrm {I}}(u_1,u_1)=2H,\ {\mathrm {I}}\!{\mathrm {I}}(u_1,u_2)=1,\ \ {\mathrm {I}}\!{\mathrm {I}}(u_2,u_2)=0.\) 
Next we see the case \(\psi \) 
is harmonic.
Using (REF ) and (REF )
again, we have
\(R_{3113}=-7 \cos ^{2}\mu +\sin ^{2}\mu ,\ \ R_{3223}=-7 \sin ^{2}\mu +\cos ^{2}\mu .\) 
Thus \(R_{3113}=R_{3223}\)  if only if
\(\mu =\pm \pi /4\) . Without
loss of generality, we may assume
\(\mu =\pi /4\) .
In this case the principal
frame \(\lbrace e_1,e_2\rbrace \)  is given by
\(e_1=\frac{1}{\sqrt{2}}(u_1+u_2),\ \ e_2=\frac{1}{\sqrt{2}}(-u_1+u_2).\) 
By definition, \({\mathrm {I}}\!{\mathrm {I}}(e_1,e_2)=0\) .
On the other hand, direct computation using
(REF ) shows
\({\mathrm {I}}\!{\mathrm {I}}(e_1,e_2)=-H\) . Thus
a constant mean curvature
surface \(M\)  with \(c=0\) 
satisfying
\(R_{3113}=R_{3223}\)  is minimal.

